There a variety of things to consider about how the project will be managed, by whom and to what level of detail.
Supplier does all the project management.
Done solely by the client or jointly client and supplier, perhaps at different levels of granularity and focus. Needs clarity of roles and resposibilities.
Either approach still benefits from clearly having the roles and responsibilities documented and agreed. It may be worth considering:
To what depth and formality a plan needs to be defined and documented and who is accountable for doing it. Describing the phases of the project, clarity of terminology, a definition of "done", relationship between tasks, identification of critical path. Does it need a documented plan and a schedule (Gantt chart). What format should it be provided in: Microsoft Project, Excel, online etc How will it be tracked and changed, and identified as a specific version of the plan.
How will project progress be measured? Time elapsed, hours expended or items delivered. What level of reports and updates will be given, daily? weekly or at key milestones, and in what format verbal, email, written report, and the level of data and interpretation to be provided. hat is the definition for Red/Amber/Green (RAG) reporting. Is there a need for a forecast for updated cost/delivery date.
Who will control the various sub tasks and items in the project. Is there a level of tolerance that is acceptable? and if the project goes out of agreed tolerance levels how is it escalated?